1/(1-x)^2从0到2的广义积分怎么求,结果是-2还是发散,我求的-2答案是发散,哪个才对?

Question

1/(1-x)^2从0到2的广义积分怎么求,结果是-2还是发散,我求的-2答案是发散,哪个才对?

Answer 1

结果是发散的
∫(0,2) dx/(x-1)²
= ∫(0,1) dx/(x-1)² + ∫(1,2) dx/(x-1)²
= ∫(0,1) d(x-1)/(x-1)² + ∫(1,2) d(x-1)/(x-1)
= 1/(1-x) [0,1] + 1/(1-x) [1,2]
= [1/(1-1) - 1/(1-0)] + [1/(1-2) - 1/(1-1)]
= 1/0 - 1/0 - 2
= +∞